9.4 Decision-Feedback Differential Detection in Fading Channels

9.4 Decision-Feedback Differential Detection in Fading Channels

9.4.1 Decision-Feedback Differential Detection in Flat-Fading Channels

The coherent detection methods discussed in Section 9.3 require explicit or implicit estimation of the fading channel, which in turn requires the transmission of pilot or training symbols. In this section we discuss decision-feedback differential detection in flat-fading channels, which does not require channel estimation. Consider again the signal model (9.19). Assume that the transmitted symbols { b [ i ]} are the outputs of a differential encoder:

Equation 9.49

where { d [ i ]} is a sequence of PSK information symbols. In simple differential detection, the complex plane is divided into M disjoint sectors, where M is the size of the PSK signaling alphabet. The detected information symbol is determined by the sector into which the complex number ( r [ i ] r [ i “ 1] ^* ) falls . Such a simple differential detection rule incurs a 3 dB performance loss compared with coherent detection in AWGN channels [396]. In flat-fading channels, however, it exhibits an irreducible error floor in the high-SNR region [222]. For example, for binary DPSK we have

Equation 9.50

where r is the correlation coefficient between the fading gains at two consecutive symbol intervals.

Multiple-symbol decision-feedback differential detection was developed in [441]. This method makes use of the correlation function of the channel and can significantly reduce the error floor of simple differential detection. In multiple-symbol differential detection [101, 102, 179, 304], an observation interval of length N is introduced. Define the following quantities :

Similar to (9.24)-(9.25), we can write the log- likelihood function as

Equation 9.51

where

Equation 9.52

Equation 9.53

The maximum-likelihood decision metric thus becomes

Equation 9.54

Since T ^-1 is symmetric, so is T . That is, if we denote T = [ t _i,j ], then t _i,j = t _j,i . Hence we can write

Equation 9.55

where (9.55) follows from (9.49). In decision-feedback differential detection, the previous information symbols d [ i - 1], d [ i - 2], ..., d [ i - N + 2] in (9.55) are assumed to take values given by the previous decisions (i.e., , and we will make a decision on the current information symbol d [ i ] to minimize the cost function r ( d [ i ]) above. To that end, such a decision rule can be simplified to [441]

Equation 9.56

Based on (9.56), we arrive at the following decision rule: Divide the complex plane into M disjoint sectors and determine by the sector into which the complex number

Equation 9.57

falls. The multiple-symbol decision-feedback differential detection algorithm is summarized as follows.

Algorithm 9.2: [Multiple-symbol decision-feedback differential detection] Given the decision memory order N , fading statistics S _N , and signal-to-noise ratio E _s / s ² :

• Compute the feedback filter coefficients from .

• Estimate the initial information symbols by simple differential detection .

• For i = N, N + 1, ..., estimate according to (9.55) .

The corresponding multiple-symbol decision-feedback differential receiver structure is shown in Fig. 9.1, where the coefficients of the feedback filter are given by the metric coefficients t _j = t , _j , 1 j N - 1. Note that when N = 2, this receiver reduces to the simple differential detector.

Figure 9.1. Structure of a multiple-symbol decision-feedback differential detector.

Simulation Examples

For all simulation results presented below, a differential QPSK constellation is used. The feedback metric coefficients are obtained from the sample autocorrelation of the simulated fading process. Figures 9.2, 9.3, and 9.4 show the BER performance of the decision-feedback differential detector in flat-fading channels with normalized Doppler frequencies B _d T equal to 0.003, 0.0075, and 0.01, respectively. It is seen that the error floors of the simple differential detector ( N = 2) are reduced by the decision-feedback differential detector (DF-DD) with N = 3 and N = 4.

Figure 9.2. BER performance of a decision-feedback differential detector in a flat-fading channel with B _d T = 0.003.

Figure 9.3. BER performance of a decision-feedback differential detector in a flat-fading channel with B _d T = 0.0075.

Figure 9.4. BER performance of a decision-feedback differential detector in a flat-fading channel with B _d T = 0.01.

9.4.2 Decision-Feedback Space-Time Differential Decoding

In what follows we extend the decision-feedback differential detection method to systems employing multiple transmit antennas and space-time differential block coding, and develop a decision-feedback space-time differential decoding technique for flat-fading channels. The material presented here was developed in [282].

Space-Time Differential Block Coding

Space-time differential block coding was developed in [204, 474]. Consider a communication system with two transmit antennas and one receive antenna. Let the information PSK symbols at time i be

Define the following matrices:

Equation 9.58

It is easy to see that G [ i ] is an orthogonal matrix (i.e., G [ i ] G [ i ] ^H = G [ i ] ^H G [ i ] = I ₂ ). The space-time differential block code is recursively defined as follows. Denote

Equation 9.59

where x [0], x [1] A . Then define X [ i ] as

Equation 9.60

By simple induction, it is easy to show that the matrix X [ i ] has the form

Equation 9.61

Hence X [ i ] is also an orthogonal matrix, and by (9.60), we have

Equation 9.62

At time slot 2 i , the symbols on the first row of X [ i ], x [2 i ] and x [2 i + 1], are transmitted simultaneously from antenna 1 and antenna 2, respectively. At time slot 2 i + 1, the symbols on the second row of X [ i ], - x [2 i + 1]* and x [2 i ]*, are transmitted simultaneously from the two antennas. We first consider the case where the channel is static. Let a ₁ and a ₂ be the respective complex fading gains between the two transmit antennas and the receive antenna. The received signals in time slots 2 i and 2 i + 1 are given, respectively, by

Equation 9.63

Equation 9.64

where n [2 i ] and n [2 i + 1] are independent complex Gaussian noise samples. Note that from (9.63) and (9.64), in the absence of noise, we can write the following:

Equation 9.65

Since

Equation 9.66

using (9.65) and (9.62) we have

Equation 9.67

Based on the discussion above, we arrive at the following differential space-time decoding algorithm.

Algorithm 9.3: [Differential space-time decoding] Form Y [0] according to (9.65) using y [0] and y [1]. For i = 1,2,...:

• Form the matrix Y [ i ] according to (9.65) using y [2 i ] and y [2 i + 1].

• Obtain an estimate of G [ i ] that is closest to Y [ i ] ^H Y [ i - 1].

Decision-Feedback Space-Time Differential Decoding in Flat-Fading Channels

We now consider decoding of the space-time differential block code in flat-fading channels. In such channels the received signals become

Equation 9.68

Equation 9.69

where { a ₁ [ i ]} and { a ₂ [ i ]} are the fading processes associated with the channels between the two transmit antennas and the receive antenna, which as before, are modeled as mutually independent complex Gaussian processes with Jakes' correlation structure. To simplify the receiver structure, we make the assumption that the channels remain constant over two consecutive symbol intervals (i.e., a ₁ [2 i ] = a ₁ [2 i + 1] and a ₂ [2 i ] = a ₂ [2 i + 1]). Then (9.68) and (9.69) can be written as

Equation 9.70

As before, denote

Then we have

Equation 9.71

The conditional log-likelihood function is given by

Equation 9.72

where

Equation 9.73

where denotes the Kronecker matrix product. Note that X [ i ][ X [ i ] ^H = X [ i ] ^H X [ i ] = I _{2 N} , and hence we have

Equation 9.74

Equation 9.75

As before, denote ; then we have

Equation 9.76

The maximum-likelihood decoding metric becomes

Equation 9.77

Note that since T = [ t _ij ] is symmetric, the cost function r ( G _n ) above can be written as

Equation 9.78

where (9.78) follows from (9.60). Since the first term in (9.78) is independent of G [ i ], the decision rule (9.77) becomes

Equation 9.79

Finally, by replacing the previous symbol matrices G [ i - 1], . . ., G [ i - N + 2] by their estimates , we obtain the following decision-feedback decoding rule for G [ i ]:

Equation 9.80

The decision-feedback space-time differential decoding algorithm is summarized as follows.

Algorithm 9.4: [Multiple-symbol decision-feedback space-time differential decoding]

• Based on the decision memory order N , the fading statistics S _N , and the signal-to-noise ratio E _s / s ² , compute the feedback metric coefficients from T = ( S _N + ( s ² I _N / E _s )) ^-1 .

• Estimate the initial symbol matrices: for i = 1,2, . . ., N - 1, estimate simply by quantizing Y [ i ] ^H Y [ i - 1].

• For i = N, N + 1, . . ., estimate according to (9.80).

The structure of a decision-feedback space-time differential decoder is shown in Fig. 9.5.

Figure 9.5. Structure of a decision-feedback space-time differential decoder.

Simulation Examples

Assume two transmit antennas and one receive antenna. By assuming that the fading process remains constant over the duration of two symbol intervals, Figs. 9.6, 9.7, and 9.8 show the BER performance of the decision-feedback space-time differential decoder in flat-fading channels with normalized Doppler B _d T = 0.003, 0.0075, and 0.01, respectively. The performance of a single-antenna system is also shown. It is seen that space-time coding provides diversity gains over single-antenna systems. Moreover, the multiple-symbol decision-feedback decoding scheme reduces the error floor exhibited by the simple space-time differential decoding method in fading channels. Although the multiple-symbol decoding scheme above is derived based on the assumption that the fading remains constant over two consecutive symbols, little performance degradation is incurred when the channels actually vary from symbol to symbol. This is illustrated in Figs. 9.9, 9.10, and 9.11, where the simulation conditions are the same as before except that the fading processes now vary from symbol to symbol. It is seen that the performance degradation due to such a modeling mismatch is negligible for practical Doppler frequencies.

Figure 9.6. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.003. Channels vary every two symbols.

Figure 9.7. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.0075. Channels vary every two symbols.

Figure 9.8. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.01. Channels vary every two symbols.

Figure 9.9. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.003. Channels vary every symbol.

Figure 9.10. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.0075. Channels vary every symbol.

Figure 9.11. BER performance of decision-feedback space-time differential decoding in flat-fading channels with normalized Doppler B _d T = 0.01. Channels vary every symbol.